Casson towers and filtrations of the smooth knot concordance group

Abstract

The n-solvable filtration $\{{\mathcal{F}}_n{\}}_{n=0}^{\infty }$ of the smooth knot concordance group (denoted by $\mathcal{C}$), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of $\mathcal{C}$ due to Cochran-Harvey-Horn, the positive and negative filtrations, denoted by $\{{\mathcal{P}}_n{\}}_{n=0}^{\infty }$ and $\{{\mathcal{N}}_n{\}}_{n=0}^{\infty }$ respectively. In particular, we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball with only positive (resp. negative) kinks in the base-level kinky disk, then K is in ${\mathcal{P}}_n$ (resp. ${\mathcal{N}}_n$). En route to this result we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball, it bounds an embedded (symmetric) grope of height n+2, and is therefore, n-solvable (this also implies that topologically slice knots bound arbitrarily tall gropes in the 4-ball). We also define a variant of Casson towers and show that if K bounds a tower of type (2,n) in the 4-ball, it is n-solvable. If K bounds such a tower with only positive (resp. negative) kinks in the base-level kinky disk then K is in ${\mathcal{P}}_n$ (resp. ${\mathcal{N}}_n$). Our results show that either every knot which bounds a Casson tower of height three is topologically slice or there exists a knot which is not topologically slice but lies in each ${\mathcal{F}}_n$. We also give a 3-dimensional characterization, up to concordance, of knots which bound kinky disks in the 4-ball with only positive (resp. negative) kinks; such knots form a subset of ${\mathcal{P}}_0$ (resp. ${\mathcal{N}}_0$).

30 pages, 21 figures; version 2 has 34 pages and 22 figures, more detailed discussion and better exposition at several places due to comments from an anonymous referee, added the word `smooth' in the title, to appear in Algebraic & Geometric Topology